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    "尽管强大的深度学习框架可以减少大量重复性工作，但若过于依赖它提供的便利，会导致我们很难深入理解深度学习是如何工作的。因此，本节将介绍如何只利用NDArray和autograd来实现一个线性回归的训练。\n",
    "\n",
    "首先，导入本节中实验所需的包或模块，其中的matplotlib包可用于作图，且设置成嵌入显示。"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "origin_pos": 2,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "from d2l import torch as d2l\n",
    "import torch\n",
    "import random"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 4
   },
   "source": [
    "## 生成数据集\n",
    "\n",
    "我们构造一个简单的人工训练数据集，它可以使我们能够直观比较学到的参数和真实的模型参数的区别。设训练数据集样本数为1000，输入个数（特征数）为2。给定随机生成的批量样本特征 $\\mathbf{X}\\in \\mathbb{R}^{1000 \\times 2}$. ，我们使用线性回归模型真实权重 $\\mathbf{w} = [2, -3.4]^\\top$ 和偏差 $b = 4.2$ ，以及一个随机噪声项 ϵ 来生成标签\n",
    "\n",
    "$$\\mathbf{y}= \\mathbf{X} \\mathbf{w} + b + \\mathbf\\epsilon.$$\n",
    " \n",
    "其中噪声项 ϵ 服从均值为0、标准差为0.01的正态分布。噪声代表了数据集中无意义的干扰。下面，让我们生成数据集\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "origin_pos": 5,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def synthetic_data(w, b, num_examples):  #@save\n",
    "    \"\"\"Generate y = Xw + b + noise.\"\"\"\n",
    "    X = torch.normal(0, 1, (num_examples, len(w)))\n",
    "    y = torch.matmul(X, w) + b\n",
    "    y += torch.normal(0, 0.01, y.shape)\n",
    "    return X, y.reshape((-1, 1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "origin_pos": 7,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "true_w = torch.tensor([2, -3.4])\n",
    "true_b = 4.2\n",
    "features, labels = synthetic_data(true_w, true_b, 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 8
   },
   "source": [
    "注意，features的每一行是一个长度为2的向量，而labels的每一行是一个长度为1的向量（标量）\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "origin_pos": 9,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "features: tensor([1.0433, 0.2341]) \n",
      "label: tensor([5.4905])\n"
     ]
    }
   ],
   "source": [
    "print('features:', features[0],'\\nlabel:', labels[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 10
   },
   "source": [
    "通过生成第二个特征`features[:, 1]`和标签 labels 的散点图，可以更直观地观察两者间的线性关系，我们将plt作图函数以及use_svg_display函数和set_figsize函数定义在d2lzh包里。以后在作图时，我们将直接调用d2lzh.plt。由于plt在d2lzh包中是一个全局变量，我们在作图前只需要调用d2lzh.set_figsize()即可打印矢量图并设置图的尺寸"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "origin_pos": 11,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
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   "source": [
    "d2l.set_figsize()\n",
    "# The semicolon is for displaying the plot only\n",
    "d2l.plt.scatter(d2l.numpy(features[:, 1]), d2l.numpy(labels), 1);"
   ]
  },
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   "metadata": {
    "origin_pos": 12
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   "source": [
    "## 读取数据集\n",
    "\n",
    "R在训练模型的时候，我们需要遍历数据集并不断读取小批量数据样本。这里我们定义一个函数：它每次返回batch_size（批量大小）个随机样本的特征和标签\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "origin_pos": 13,
    "tab": [
     "pytorch"
    ]
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   "outputs": [],
   "source": [
    "def data_iter(batch_size, features, labels):\n",
    "    num_examples = len(features)\n",
    "    indices = list(range(num_examples))\n",
    "    # The examples are read at random, in no particular order\n",
    "    random.shuffle(indices)\n",
    "    for i in range(0, num_examples, batch_size):\n",
    "        batch_indices = torch.tensor(\n",
    "            indices[i: min(i + batch_size, num_examples)])\n",
    "        yield features[batch_indices], labels[batch_indices]"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 15
   },
   "source": [
    "通常，请注意，我们要使用大小合适的微型批处理来利用GPU硬件，该硬件擅长并行化操作。因为每个示例都可以并行地通过我们的模型进行馈送，并且每个示例的损失函数的梯度也可以并行获取，所以GPU允许我们在处理数百个示例时所花的时间几乎比处理单个示例所花费的时间要多\n",
    "\n",
    "让我们读取第一个小批量数据样本并打印。每个批量的特征形状为(10, 2)，分别对应批量大小和输入个数；标签形状为批量大小.\n"
   ]
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "origin_pos": 16,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "tensor([[ 1.2663,  0.8296],\n",
      "        [ 0.3822, -0.2893],\n",
      "        [-1.0795, -0.4111],\n",
      "        [ 1.2304,  1.6950],\n",
      "        [-0.5994,  1.5374],\n",
      "        [-2.1674, -0.2711],\n",
      "        [-0.1666,  1.1178],\n",
      "        [ 0.7400, -2.2019],\n",
      "        [ 1.3277,  0.4184],\n",
      "        [-0.3571, -0.0699]]) \n",
      " tensor([[ 3.9030],\n",
      "        [ 5.9481],\n",
      "        [ 3.4511],\n",
      "        [ 0.9004],\n",
      "        [-2.2246],\n",
      "        [ 0.7888],\n",
      "        [ 0.0572],\n",
      "        [13.1685],\n",
      "        [ 5.4582],\n",
      "        [ 3.7180]])\n"
     ]
    }
   ],
   "source": [
    "batch_size = 10\n",
    "\n",
    "for X, y in data_iter(batch_size, features, labels):\n",
    "    print(X, '\\n', y)\n",
    "    break"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 17
   },
   "source": [
    "在运行迭代时，我们会依次获取不同的迷你批，直到耗尽整个数据集为止（尝试此操作）。尽管上面实现的迭代对于教学目的是有好处的，但它的效率低下可能会使我们在实际问题上遇到麻烦。例如，它要求我们将所有数据加载到内存中，并执行大量随机内存访问。深度学习框架中实现的内置迭代器效率更高，并且可以处理存储在文件中的数据和通过数据流馈送的数据\n",
    "\n",
    "## 初始化模型参数\n",
    "\n",
    "我们将权重初始化成均值为0、标准差为0.01的正态随机数，偏差则初始化成0\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "origin_pos": 19,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "w = torch.normal(0, 0.01, size=(2,1), requires_grad=True)\n",
    "b = torch.zeros(1, requires_grad=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 21
   },
   "source": [
    "初始化参数后，我们的下一个任务是更新它们，直到它们足够适合我们的数据为止。每次更新都需要考虑我们的损失函数相对于参数的梯度。给定该梯度，我们可以在可能减少损耗的方向上更新每个参数。\n",
    "\n",
    "由于没有人想要显式地计算梯度（这很繁琐且容易出错），因此我们使用自动微分（如 2.5节所述）来计算梯度.\n",
    "\n",
    "\n",
    "## 定义模型\n",
    "\n",
    "接下来，我们必须定义模型，将其输入和参数与其输出相关联。回想一下，要计算线性模型的输出，我们只需获取输入要素的矩阵矢量点积  X  和模型权重  w ，然后加上偏移量  b 每个示例。注意下面 Xw  是向量，  b 是一个标量。回顾第2.1.3节中描述的广播机制。当我们添加一个向量和一个标量时，该标量被添加到向量的每个分量上.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "origin_pos": 22,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def linreg(X, w, b):  #@save\n",
    "    \"\"\"The linear regression model.\"\"\"\n",
    "    return torch.matmul(X, w) + b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 23
   },
   "source": [
    "## 定义损失函数\n",
    "\n",
    "由于更新模型需要采用损失函数的梯度，因此我们应该首先定义损失函数。在这里，我们将使用3.1节中描述的平方损失函数 。在实现中，我们需要将真实值y转换为预测值的形状 y_hat。以下函数返回的结果的形状也将与相同y_hat\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "origin_pos": 24,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def squared_loss(y_hat, y):  #@save\n",
    "    \"\"\"Squared loss.\"\"\"\n",
    "    return (y_hat - y.reshape(y_hat.shape)) ** 2 / 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 25
   },
   "source": [
    "## 定义优化算法\n",
    "\n",
    "正如我们在第3.1节中讨论的那样，线性回归具有封闭形式的解决方案。但是，这不是一本关于线性回归的书，而是一本关于深度学习的书。由于本书介绍的其他模型都无法解析解决，因此我们将借此机会介绍您的第一个小批量随机梯度下降的工作示例。\n",
    "\n",
    "在每一步中，使用从数据集中随机抽取的一个小批量，我们将估计损耗相对于参数的梯度。接下来，我们将朝着减少损耗的方向更新参数。给定一组参数，学习率和批次大小，以下代码将应用minibatch随机梯度下降更新。更新步骤的大小取决于学习率 lr。因为我们的损失是根据示例的最小批量计算得出的，所以我们通过批量大小（batch_size）对步长进行归一化，因此典型步长的大小在很大程度上不取决于我们对批量大小的选择\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "origin_pos": 27,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [],
   "source": [
    "def sgd(params, lr, batch_size):  #@save\n",
    "    \"\"\"Minibatch stochastic gradient descent.\"\"\"\n",
    "    for param in params:\n",
    "        param.data.sub_(lr*param.grad/batch_size)\n",
    "        param.grad.data.zero_()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 29
   },
   "source": [
    "## 训练\n",
    "\n",
    "Now that we have all of the parts in place,\n",
    "we are ready to implement the main training loop.\n",
    "It is crucial that you understand this code\n",
    "because you will see nearly identical training loops\n",
    "over and over again throughout your career in deep learning.\n",
    "\n",
    "In each iteration, we will grab a minibatch of training examples,\n",
    "and pass them through our model to obtain a set of predictions.\n",
    "After calculating the loss, we initiate the backwards pass through the network,\n",
    "storing the gradients with respect to each parameter.\n",
    "Finally, we will call the optimization algorithm `sgd`\n",
    "to update the model parameters.\n",
    "\n",
    "In summary, we will execute the following loop:\n",
    "\n",
    "* Initialize parameters $(\\mathbf{w}, b)$\n",
    "* Repeat until done\n",
    "    * Compute gradient $\\mathbf{g} \\leftarrow \\partial_{(\\mathbf{w},b)} \\frac{1}{|\\mathcal{B}|} \\sum_{i \\in \\mathcal{B}} l(\\mathbf{x}^{(i)}, y^{(i)}, \\mathbf{w}, b)$\n",
    "    * Update parameters $(\\mathbf{w}, b) \\leftarrow (\\mathbf{w}, b) - \\eta \\mathbf{g}$\n",
    "\n",
    "In each *epoch*,\n",
    "we will iterate through the entire dataset\n",
    "(using the `data_iter` function) once\n",
    "passing through every example in the training dataset\n",
    "(assuming that the number of examples is divisible by the batch size).\n",
    "The number of epochs `num_epochs` and the learning rate `lr` are both hyperparameters,\n",
    "which we set here to 3 and 0.03, respectively.\n",
    "Unfortunately, setting hyperparameters is tricky\n",
    "and requires some adjustment by trial and error.\n",
    "We elide these details for now but revise them\n",
    "later in\n",
    ":numref:`chap_optimization`.\n"
   ]
  },
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   "execution_count": 12,
   "metadata": {
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   "source": [
    "lr = 0.03\n",
    "num_epochs = 3\n",
    "net = linreg\n",
    "loss = squared_loss"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "origin_pos": 32,
    "tab": [
     "pytorch"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch 1, loss 0.036363\n",
      "epoch 2, loss 0.000132\n",
      "epoch 3, loss 0.000053\n"
     ]
    }
   ],
   "source": [
    "for epoch in range(num_epochs):\n",
    "    for X, y in data_iter(batch_size, features, labels):\n",
    "        l = loss(net(X, w, b), y)  # Minibatch loss in `X` and `y`\n",
    "        # Compute gradient on `l` with respect to [`w`, `b`]\n",
    "        l.sum().backward()\n",
    "        sgd([w, b], lr, batch_size)  # Update parameters using their gradient\n",
    "    with torch.no_grad():\n",
    "        train_l = loss(net(features, w, b), labels)\n",
    "        print(f'epoch {epoch + 1}, loss {float(train_l.mean()):f}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 34
   },
   "source": [
    "In this case, because we synthesized the dataset ourselves,\n",
    "we know precisely what the true parameters are.\n",
    "Thus, we can evaluate our success in training\n",
    "by comparing the true parameters\n",
    "with those that we learned through our training loop.\n",
    "Indeed they turn out to be very close to each other.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "origin_pos": 35,
    "tab": [
     "pytorch"
    ]
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "error in estimating w: tensor([ 0.0004, -0.0007], grad_fn=<SubBackward0>)\n",
      "error in estimating b: tensor([0.0007], grad_fn=<RsubBackward1>)\n"
     ]
    }
   ],
   "source": [
    "print(f'error in estimating w: {true_w - w.reshape(true_w.shape)}')\n",
    "print(f'error in estimating b: {true_b - b}')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 36
   },
   "source": [
    "Note that we should not take it for granted\n",
    "that we are able to recover the parameters perfectly.\n",
    "However, in machine learning, we are typically less concerned\n",
    "with recovering true underlying parameters,\n",
    "and more concerned with parameters that lead to highly accurate prediction.\n",
    "Fortunately, even on difficult optimization problems,\n",
    "stochastic gradient descent can often find remarkably good solutions,\n",
    "owing partly to the fact that, for deep networks,\n",
    "there exist many configurations of the parameters\n",
    "that lead to highly accurate prediction.\n",
    "\n",
    "\n",
    "## Summary\n",
    "\n",
    "* We saw how a deep network can be implemented and optimized from scratch, using just tensors and auto differentiation, without any need for defining layers or fancy optimizers.\n",
    "* This section only scratches the surface of what is possible. In the following sections, we will describe additional models based on the concepts that we have just introduced and learn how to implement them more concisely.\n",
    "\n",
    "\n",
    "## Exercises\n",
    "\n",
    "1. What would happen if we were to initialize the weights to zero. Would the algorithm still work?\n",
    "1. Assume that you are\n",
    "   [Georg Simon Ohm](https://en.wikipedia.org/wiki/Georg_Ohm) trying to come up\n",
    "   with a model between voltage and current. Can you use auto differentiation to learn the parameters of your model?\n",
    "1. Can you use [Planck's Law](https://en.wikipedia.org/wiki/Planck%27s_law) to determine the temperature of an object using spectral energy density?\n",
    "1. What are the problems you might encounter if you wanted to  compute the second derivatives? How would you fix them?\n",
    "1.  Why is the `reshape` function needed in the `squared_loss` function?\n",
    "1. Experiment using different learning rates to find out how fast the loss function value drops.\n",
    "1. If the number of examples cannot be divided by the batch size, what happens to the `data_iter` function's behavior?\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "origin_pos": 38,
    "tab": [
     "pytorch"
    ]
   },
   "source": [
    "[Discussions](https://discuss.d2l.ai/t/43)\n"
   ]
  }
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